Maze vs labyrinth
A maze is corridors with junctions. You choose, sometimes wrongly, and turn around. That choice is the puzzle.
A labyrinth — Chartres pavement, the Theseus story — is often a single winding path. You cannot get lost if you keep walking. On this site the split is strict: if there is a choice, it is a maze.
What “perfect” means
Between any two rooms there is exactly one route. No loops, no walled-off islands. Room count R, open doorways R−1. Same arithmetic as a tree.
Fog and keys do not add loops. Fog hides. Keys sit in rooms. The exit is still on the unique spanning tree. Circle, triangle, and hex outlines follow the same contract on their cell graphs.
Why the words get mixed
Tourist signs and puzzle books often say labyrinth for both objects. The maze section keeps the branching word for the game: wrong turns exist, and the right-hand rule depends on that simply connected layout.